Resumen de On the local-global principle for isogenies of abelian surfaces

Davide Lombardo, Matteo Verzobio

• Let ɭ a prime number. We classify the subgroups G of Sp4(F) and GSp4(F) that act irreducibly on F4 , but such that every element of G fixes an F-vector subspace of dimension 1. We use this classification to prove that a local-global principle for isogenies of degree ɭ between abelian surfaces over number fields holds in many cases—in particular, whenever the abelian surface has non-trivial endomorphisms and ɭ is large enough with respect to the field of definition. Finally, we prove that there exist arbitrarily large primes ɭ for which some abelian surface A/Q fails the local-global principle for isogenies of degree ɭ.